DOUBLY-EXCITED STATES OF THE ALUMINIUM ISOELECTRONIC SEQUENCE

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DOUBLY-EXCITED STATES OF THE ALUMINIUM ISOELECTRONIC SEQUENCE

R. Crossley

To cite this version:

R. Crossley. DOUBLY-EXCITED STATES OF THE ALUMINIUM ISOELECTRONIC SEQUENCE.

Journal de Physique Colloques, 1970, 31 (C4), pp.C4-155-C4-159. �10.1051/jphyscol:1970425�. �jpa-

00213880�

DOUBLY-EXCITED STATES OF THE ALUMINIUM ISOELECTRONIC SEQUENCE

by R . CROSSLEY

Department of Mathematics, University of York, York, England

Rksumk. - On discute la methode de I'expansion a la charge nucleaire Z pour effectuer le calcul d'energies des ions atoniiques qui font partie d'un ordre isoCIectronique donnC i I'tgard des niethodes d'experience et d'autre calcul. On se servit de la niethode pour examiner I'interaction de configurations entre les ternies 'D, 'DO et 'PO de I'ordre isoelectronique d'aluminiurn. On propose quelques revisions des analyses acceptees.

Abstract.

^-

The Z-expansion method for calculating the energies of atomic ions belonging to a given isoelectronic sequence is discussed in relation to experimental and other theoretical methods.

The method is used to investigate configuration interaction between ID,

2DO

and

2P'J

terms of aluminium-like ions. Revisions of the accepted analyses are suggested.

I. Introduction.

-

Over the last twenty years developments in astrophysics and plasma physics liave lead t o a renewal of interest in atomic spectroscopy.

Most previous work in this field, whether by experi- ment o r calculation, was confined to the study of neutral atoms and systems of a very few degrees of ionization. T h e realisation that an understanding of basic processes in the sun alone r e q ~ ~ i r e d knowledge of such ions as S i y + and Fel"+ is suecient indication of the extent of the problem. For the theoretician two approaches were possible

^-

either to extend the classical methods of c a l c ~ ~ l a t i o n (as described by e. g.

Slater [I]) o r to develop new methods appropriate t o these new problems. Into tlie latter category falls the Z-expansion method, introduced for Iieli~~m-like systems by Hylleraas [2] (see also Bethe and Salpe- ter 131 pp. 151-3) and developed for 11101-e complex atoms by Layzer [4], Linderberg and Shull [5] and Crossley and Coulson [6]. Extensive calc~llations by this metliod liave been carried out by Godfredsen [7], but recently a number of a~itliors [g-l 1] liave criti- cized the technique and preferred other ~netliods.

The purpose of the present contribution is to attempt to answer some of these criticisms and to point out the particular advantages of tlie 2-expansion method.

We find it convenient to illustrate our arguments by reference to tlie aluminium isoelectt-onic sequence for the following reason. Classical empirical methods of spectral analysis 1121 applied to the smaller atoms are usually adequate for the interpretation of Rydberg series of levels, unless these are strongly perturbed

by 3 p2 ' D) and in the aluminium sequence (perturba- tion of the 3 sZ

11

d 2 D series by 3 s 3 p 2 'D). Levels which d o not belong to simple Rydberg series also provide proklems of analysis, and good examples of these are provided by the doubly-excited configura- tions 3 s 3 1 3 1' of the aluminium sequence.

11. Theory.

-

The Z-expansion method depends on a device introduced by Hylleraas [2]. If we write the Russell-Saunders Hamiltonian of a n N-electron atom (or ion) with nuclear charge Z in terms of a unit of length of Z a. u. and a unit of energy of Z 2 a.

U.

(i.

^{5 .}

different units for each N-electron ion) we obtain

( i , j

=

1 ,..., N ) . ₍₁₎ Dividing this Hamiltonian according t o the brackets and writing

~e

⁼

H,

-I-

Z - I H , (2) enables us to treat the problem by standard Rayleigh- Schrijdinger perturbation theory, t h e zero-order problem being

a n d Z-' being a natural expansion parameter. The resulting expression for the total energy E is (in the original atomic units)

by configuration interaction. T h e classical examples of E

=

Eo 2' + El Z + Ez + E3

^{Z - I}

+ -..

such perturbation [13] occur in the magnesium ⁽⁴⁾

sequence ('1 (perturbation o f the 3 s d ' D series where the coefficients E, are constant for a given iso-

(1)

The magnesium

^has

been studied by

a

number electronic sequence (given N) since Ho and H I are of authors

{8-11, 141, and

see

$ I V below.

both independent of Z .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970425

C4- 156

R. CROSSLEY

The zero-order problem (3) is easily solved since Ho involving no change of principal quantum number, and is a sum of hydrogenic Hamiltonians. Thus considering a plot of excitation energy against Z, E, is the asymptotic gradient of the curve. Further.

1 1

Eo

= -

-1

;

(5) cases involving change of principal quantum numbel

2

^{i 11;}

can be reduced to this case since the dependence on Z-'

is trivially calculated from (5). Now each value of E , where the n i are the principal quantum numbers of the

(each gradient of the excitation energy curve) is a hydrogenic spin-orbitals appearing in the correspond-

difference of eigenvalues of the interaction matrices ing eigenfunction

t j o

(most generally a linear combina-

of the first-order degenerate perturbation theory.

tion of Slater determinants).

The corresponding eigenvectors are the coefficients 01 first-order energy

E l

is obtained tile

⁽⁽

correct

⁾⁾

zero-order wave-functions of t h e standard methods of degenerate perturbation

and lower states of the transition as lineal.

Since E~ depends

O n { I z i

1 'eq. 5, the combinatiolls of single-col~fig~~ratioll wave-fi~nctions degeneracy can be [71

^'Ias

These

s

mixing)) coefficients may be calculated to an) carried out calcr~lations of E , for a wide range of desired accuracy and are independent of Z. Thc atomic taking the

^{@ 0}

as sin%1e 'Iater determi- as explaiIled above, requires cel-tain configLl.

nants

;

consequently in each calc'latioll he 'as 'ad

^'0

rations to be illcluded i n the calculations

I l o

ot,zel..r allow for degeneracy between all configurations Thus the theory confirms the idea of associating (defined by

^{* i ' 1}

1) ^the

⁽⁽

') [41 energy levels along an isoelectronic sequence, and give.

defined by { ni ] and parity. The degeneracy can be each sequence of energy levels a unique multi-confl- reduced by using zero-order wave-functions of defi- gllratiorlal

nite total symmetry

;

then it will only be necessary to

include configurations from the complex (

11,

j- which lV. Relation to other theoretical metltods.

^-

yield terms of the symmetry of interest ('). Most other methods for calculating energy levels

ol

Calculations of higher-order energy coefficients complex atoms depend on the variational principle [ I ] have been carried

O u t

(a brief survey is given by _1, theory the exact wave-function could be obtainecl crossley [15]), but they will not be used in the sue- as all in terlns of some complete infinitt ceeding sections. set of basis functions, but in practice only a finiti

calculation is possible and it is necessary to seleci 111. Relation to standard methods of spectral

aria-

from the basis set those functions wl~icli (hopefi~lly lysis.

-

EdlCn [I21 describes the standard methods of are most important in lowering the energy. The basic analysis for the spectra of isoelectronic sequences. functions may be fi~nctions of a most general kind The two basic empirical laws state that the excitation (e. g. those of Hylleraas [17]), but for atoms wit11 energy of a given transition along the sequence tends more than two electrons the trial function is usuall!

to depend linearly on Z 2 increases (Moseley's a linear combination of functions each of which ma!

law

( 3 ) ) ,

but if the transition involves no change of be interpreted physically as representing an electronic

principal quantum number the dependence is Propor- configuration

;

hence the name

(<

configuration intes- tional to Z (the irregular doublet law). action

⁾⁾

(c. f. the discussion of Green et al. [18]).

Eq. (4) represents the energy of any atomic state, The single-configuration functions may of courst so it follows that any excitation ellergY is given by a contain variational parameters. Calculations of thi?

similar expression. Hence our theory gives an imme- kind have been carried out using simple Slater-type diate confirmation of Moseley's law, and we need spin-orbitals (Jucys et 01. [19] give a discussion).

only note the form of eq. (5) to obtain confirmation o f using the Hartree-Fock rnethod [15], and using the the irregular doublet law also. This immediate connec- T~lomas-Ferlni method [ I 11. The result is that tion with empirical methods is a particularly pleasing energy level is described as a linear combination ol feature of Z-expansion theory which should allay single-configuration wave-functions, but in contra- fears as to the convergence of (4). I n fact there is distillction to the Z-expansion case the mixing coefli- absolutely no evidence that the expansion (4) is ill- cieI1ts now depend firstly on Z (since a separate calcii- behaved, althougll not surprisingly for low values of lation is necessary for each valile of Z), secondly on the higher contributions play a more important r81e the configurations chosen to appear in the calculation (see e. g. the discussion of electron affinities given by (since this choice is now arbitrary) and thirdly on the

Crossley [16]). form chosen for the single-configuration trial func-

But the theory gives us more information of a tions. Thus it is wrong t o attach any universal signi- quantitative nature. Concentrating On excitations ficance to the values of mixillg coefficients

by such methods. Zare [14], for example, introduces

(?) In computer calculations it may however be preferable

the notion of

⁽⁽

spectral purity

^D,

and wishes to label

to handle the greater degeneracy problem directly rather than

atomic energy levels by configurations listed in order of

set up the more complicated wave-functions of the symrnetrised

scheme [7, 1 I].

magnitude of their mixing coefficients. Unfortunately

( 3 ) Properly for X-ray spectra.

his mixing coefficients are a function of his method

(Hartree-Fock-Slater) and s o his nomenclature can have no universality. The kind of confusion which can arise is well illustrated by the ID terms of the Mg isoelectronic sequence. An unusual feature of this sequence is that the pert~~rbatiorl between the 3 s 3 d and 3 p2 configurations is strongest not in the neutral atom but in the first positive ion A l f . Weiss [lo]

(Hartree-Fock) agrees with Zare [I 41

(

Hal-tree-Fock- Slater) in placing the 3 p2 term lower in energy than the 3 s 3 d, in agreement with the experimental iden- tification [20]

;

however Wilson [9] (Hartt-ee-Fock configuration average) and Eissner and Nussbau- mer [I I] (variationally scaled Thomas-Ferrni) obtain the opposite result. All these authors consider the isoelectronic sequence but are obliged to calculate each ion separately

;

both Wilson and Eissner and Nussbaumer present curves showing term energy as a fi~nction of Z, and the 3 s 3 d ' D and 3 p' ' D curves cross between Al' and S i 2 + . Now nothing in q u a n t ~ ~ l n mechanics prevents us considering non- integral (and so non-physical) values of Z and it is clear that it is very convenient to d o so

;

however if we regard energies as functions of a co~itinuo~rs variable Z we must accept the consequence that in a variational calculation where the trial function is a simple linear combination of functions the non-crossing rule applies : energy curves corresponding to states of the same symmetry cannot cross. A few c a l c ~ ~ l a t i o n s for non-integral Z would readily confirm this. The whole situation is in fact very similar to that of potential curves for a diatomic molecule (see e. g. Coulson [21]).

Wilson's calculations are single-configuration, but Eissner and Nussbaumer, and by implication Weiss and Zare, run into exactly this trap. The resolution is simple ; if we wish to relate terms of different ions of a n isoelectronic sequence, then we must discard tlie notion of labelling a term with the dominant configuration of its wave-fi~nction. The Z-expansion method supplies a natural label in a n entirely consis.

tent way. Of course the zero-order wave-function of Z-expansion theory is a very poor wave-function, but perturbation theory ensures a Inore satisfac- tory energy value (knowledge of $,, suffices to determine E,,,, ,). Even s o the energy calculated to first order hardly compares with other calculations except at high degress of ionisation : but the Z- expansion approach is ideal for a semi-empirical treatment such as has been carried out for ionisation potentials and electron affinities [6, 161.

V. Configuration Interaction in Aluminium.

-

Moore 1201 lists the 3 s2 nd 'D series for Al but not the 3 s 3 p2 'D terms which may be expected t o perturb the series strongly. Accordingly the question has arisen as to where the 3 s 3 p 2 term is. From the above discussion it is clear that the question is wrongly posed ; the 3 s 3 p2 and 3 s2 3 d terms will interact strongly and we cannot expect that single-configura- tion labels will be satisfactory. This point has been

made very clearly in a most thorough discussion of the problem by Weiss [22] : nevertheless it is of some interest to pursue the original rather nai've question.

The experimental situation has been summarised by Gruzdei. [23]. Apart from a suggested identifica- tion of a difl'use f c n t ~ ~ r e in tlie c o n t i n u ~ ~ m by Gar- ton [24] which was later withdrawn [25] the most likely candidates for the perturber have been thought to be the terms identified by Moore as the first three series members 3 d , 4 d , 5 d . Recent investigation of the higher series members [26] reveals no peculiarities there. Eriksson and Isberg [27] s ~ ~ g g e s t that the rut1 of quantum defects implies the perturber is 4 d o r 5 d, but their argument is unclear and they in fact retain Moore's identification

;

however Gruzdev points out that Synek's single-configuration Hartree-Fock calculations [28] support this conclusion. Gruzdev argues for 4 d fro111 the shape of the quantum defect curve and other evidence, but tlie case is unconvincing.

That the lowest 'D level is the perturber was the view of Paschen [29], and more recently Shen- stone [30, 311 has revived this proposal. Turning now to the Z-expansion method the most simple procedure is illustrated in figure I . The experimental excitation

FIG.

1.

-Term Energies

of

the Aluminium Sequence.

energies of some terms of interest are plotted and

straight lines with gradient El (from [7]) are drawn

in. In order to have a definite procedure these lines

have been drawn t h r o ~ ~ g l i the corresponding experi-

mental points of the Si+ spectrum, where we can

C4- 158

R. CROSSLEY

~~tilise the extremely thorough work of Shenstone [31] ; the rest of the experimental data is from Moore [20].

This gives a very crude empirical estimate of E,, but it is sufficient to resolve many problems of ana- lysis. Thus the 3 s 3 p2 4P and 'S terms all lie very close to their straight lines and their identification is confirmed. The ionic 3 s 3 p2 'D terms behave well ; the 3 s2 3 d 'D terms appear to diverge from their line, but study of higher ions sl~ows that the asympto- tic gradient is correct so that it seems we have here another example of the rather

⁽⁽

wild

))

behaviour of a single d-electron which has been remarked by a number of authors recently 1321. The three suspect 'D levels of A1 are indicated with asterisks. What we wish to do is to extrapolate to a lower value of Z, and it must be admitted that this is rather contrary to the nature of the Z-expansion metl~od. Never- tl~eless consideration of the 3 s 3 p2 2 D terms strongly suggests that the lowest 'D level of Alis to be corre- lated with the lowest 'D levels of the ions and so sl~ould be given the same label 3 s 3 p2 2D, or better the multi-configuration label of Godfredsen [7]

(where very small contributions from a number of other configurations which interact at zero-order have been omitted). Adopting this identification the usual quantum defect curve may be drawn for the revised 3s2 n d series. The formula for the quantum defect (see [2], p. 368) is

0.123,888

(12* - i t ) = -

0.040,899 + 0.047,972 a,, +

0 0 -

a,, and the curve is shown in figure 2. It has the normal appearance of a series perturbed from below and is in fact strikingly similar t o the corresponding curve for Sif given by Shenstone [31]. Thus from the tra- ditional standpoint a reasonable case can be made for taking Moore's 3 s2 3 d 'D term as the perturber 3 s 3 p2 2D, and the implied correlation of terms

FIG.

2. -

Quantum Defect Curve for the

3 sz 11 d

series of Alu- minium.

along the sequence agrees well with 2-expansion analysis.

There is other evidence which may be used to sup- port this identification. Of greatest interest is the consideration of the run of oscillator strengths along the diffuse series 3 p -

¹⁷

d, discussed by Penkin and Shabanova [33]. The anomalous effect they discover on the basis of Moore's identifications is reduced to the appearance of a simple configuration interaction effect with the identification suggested here. Curiously they do not discuss the 3 s 3 p2 2 D term altl~ough they do relate the anomalous behaviour to sharp fluctuations of the quantum defect [34]. Our change of identification also removes the inflections from the quantum defect curve [34, 231. Similarly it removes an anomaly from the riun of level-splittings of the

17

d 'D terms [27].

V1. Doubly-excited States of the Alurniniurn- Sequence.

-

In the same way we may use the Z- expansion technique to investigate other excited states of the A1 sequence. The configurations of greatest interest are those of the form 3 1 3 1' 3 l", and these are particularly convenient for the Z-expansion method since they all have the same set of principal q i ~ a n t ~ l r n numbers. Carrying out the same procedure leads to the following conclusions concerning the first six members of the A1 sequence (i. e. Al, S i f , P 2 + , S 3 + , c I 4 + , A 5 + ) ; experimental data is taken from Eriksson and Tsberg [27] for AI, from Shen- stone [31] for Si', and the remainder from Moore [20].

The required valrtes of E , are from Godfredsen ,[7].

a ) 3 s2 3 p, 3 s2 3 d, 3 s 3 p2

:

with the exception of the 2 D terms of Al discussed above, all identifica- tions of terms in these configurations are correct.

b) 3 p3 and 3 s 3 pt3Po) 3 d : confusions occur here because of configuration interaction in the terms 'Do and

^'PO

which arise in both config~~rations. 'Do terms identified as 3 s 3 p 3 d in A1 and Si+ should be 3 p3.

The 3 p3 2D0 terms of Sif [31] sl~ould be 3 s 3 p 3 d.

For the 'Po terms, t l ~ e 3 s 3 p 3 d of Sif S I I O L I I ~ have the same designation as the 3 p3 of P 2 + and S 3 + , but there is insufficient data to determine which identifi- cation is correct. If these are 3 p3, then the term identified as 3 p3

^*PO

i n AI' [3I] should be 3 s 3 p 3 d.

The term in P 2 + identified by Robinson [35] only as 210 is either

'PO

wit11 the same configuration as the last-mentioned term, or it c o ~ ~ l d be 3 s 3 p 3 d

2 ~ 0 .

The remaining terms of these configurations (3 p3 4S0 and 3 s 3 p 3 d 4F, D,

^PO

and 'FO) are correctly identified in the literature.

c)

The procedure can be taken further to consider

3 s2 4 d 2D. Here the A1 term appears out of place

but the term identified [27] as 3 s2 5 d 'D fits very

well. We see this result provides us with yet further

evidence in support of the changed designation of

2 D terms in A1 proposed in 6 V.

Acknowledgments.

-

I a m indebted t o Professors tion is based o n w o r k carried o u t a t H a r v a r d College A. D a l g a r n o , W. R. S . G a r t o n a n d H. A. R o b i n s o n f o r O b s e r v a t o r y with t h e s u p p o r t o f t h e N a t i o n a l Science llelpful discussions a n d correspondence. T h i s c o n t r i b u - F o u n d a t i o n .

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- 9 7

L J

I.

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(English translation

:

Opt. Spect., 1962, 12, 83.)

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:

Opt. Spect., 1966, 20, 209.)

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11,

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[27] ERIKSONS ( K .

9.

S.), ISBERG (H. B. S.), Ar.h-.

FJ's.,

1963, 23, 527.

[28] SYNEK (M.), P11y.s. Rev., 1963, 131, 1572.

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[33] PENKIN (N. P.), SHABANOVA (L. N.), Opt. Spekt., 1965, 18, 896. (English translation

: Opt. Spect.,

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:

Opt. Spect., 1965, 18, 535.)

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